Dihedral symmetry conjecture for critical points of the coupled dipolar/Maier–Saupe potential

Consider the coupled dipolar/Maier–Saupe potential with parameters $(\sigma,\kappa)$ and its critical points on the space of admissible orientational densities. A critical point has dihedral symmetry $D^2$ when it is invariant under the corresponding dihedral symmetry group. Dihedral symmetry conjecture. All the critical points have the dihedral symmetry $D^2$. This numerical conjecture would give a complete symmetry classification of the critical points for the coupled dipolar/Maier–Saupe model; no proof or resolution is supplied here.